Inharmonic timbres have a pseudo-octave — a non-2:1 interval that plays the functional role of the octave

For harmonic timbres, the octave (2:1 ratio) is the most consonant non-unison interval because the second partial of the lower note aligns exactly with the fundamental of the upper note. When a timbre has stretched partials (e.g., at 1×, 2.1×, 3.1×, …), the alignment occurs at a 2.1:1 ratio instead. This non-2:1 interval is called the pseudo-octave: it is the interval of maximum consonance for that timbre and functions as the repeating unit of the scale. Stretched piano strings naturally produce slightly stretched partials, which is one reason piano tuners stretch the tuning (octave stretch). Extreme stretching (2.2×) causes tonal fission; moderate stretching (2.1×) can still support musically coherent pieces once the ear adapts.

Examples

Sound example [S: 4] in the book: a piece played with 2.1-stretched timbres in a 2.1-stretched scale sounds musical. The same piece with 2.1-stretched timbres in standard 12-tet (no pseudo-octave) is uniformly dissonant. Octave stretch in piano tuning is a real-world application of this principle.

Assessment

A timbre has partials at 440, 924 (×2.1), 1452 (×3.3) Hz. Identify the pseudo-octave ratio. Explain why playing two notes a 2:1 ratio apart with this timbre would sound less consonant than playing two notes at a 2.1:1 ratio.